Derivative-free superiorization with component-wise perturbations

Superiorization reduces, not necessarily minimizes, the value of a target function while seeking constraints-compatibility. This is done by taking a solely feasibility-seeking algorithm, analyzing its perturbations resilience, and proactively perturbing its iterates accordingly to steer them toward a feasible point with reduced value of the target function. When the perturbation steps are computationally efficient, this enables generation of a superior result with essentially the same computational cost as that of the original feasibility-seeking algorithm. In this work, we refine previous formulations of the superiorization method to create a more general framework, enabling target function reduction steps that do not require partial derivatives of the target function. In perturbations that use partial derivatives the step-sizes in the perturbation phase of the superiorization method are chosen independently from the choice of the nonascent directions. This is no longer true when component-wise perturbations are employed. In that case, the step-sizes must be linked to the choice of the nonascent direction in every step. Besides presenting and validating these notions, we give a computational demonstration of superiorization with component-wise perturbations for a problem of computerized tomography image reconstruction.Comment: Numerical Algorithms, accepted for publicatio

Derivative-Free Superiorization: Principle and Algorithm

The superiorization methodology is intended to work with input data of constrained minimization problems, that is, a target function and a set of constraints. However, it is based on an antipodal way of thinking to what leads to constrained minimization methods. Instead of adapting unconstrained minimization algorithms to handling constraints, it adapts feasibilityseeking algorithms to reduce (not necessarily minimize) target function values. This is done by inserting target-function-reducing perturbations into a feasibility-seeking algorithm while retaining its feasibility-seeking ability and without paying a high computational price. A superiorized algorithm that employs component-wise target function reduction steps is presented. This enables derivative-free superiorization (DFS), meaning that superiorization can be applied to target functions that have no calculable partial derivatives or subgradients. The numerical behavior of our derivative-free superiorization algorithm is illustrated on a data set generated by simulating a problem of image reconstruction from projections. We present a tool (we call it a proximity-target curve) for deciding which of two iterative methods is \better" for solving a particular problem. The plots of proximity-target curves of our experiments demonstrate the advantage of the proposed derivative-free superiorization algorithm.Comment: 16 pages, one figur